Question: Simplify and expand the following expression: $ \dfrac{q}{4q + 6}+\dfrac{q}{2q - 4} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4q + 6)(2q - 4)$ Multiply the first term by $\dfrac{2q - 4}{2q - 4}$ $ \begin{align*} \dfrac{q}{4q + 6} \times \dfrac{2q - 4}{2q - 4} & = \dfrac{(q)(2q - 4)}{(4q + 6)(2q - 4)} \\ & = \dfrac{2q^2 - 4q}{(4q + 6)(2q - 4)}\end{align*} $ Multiply the second term by $\dfrac{4q + 6}{4q + 6}$ $ \begin{align*} \dfrac{q}{2q - 4} \times \dfrac{4q + 6}{4q + 6} & = \dfrac{(q)(4q + 6)}{(2q - 4)(4q + 6)} \\ & = \dfrac{4q^2 + 6q}{(2q - 4)(4q + 6)}\end{align*} $ Now we have: $ = \dfrac{2q^2 - 4q}{(4q + 6)(2q - 4)} + \dfrac{4q^2 + 6q}{(2q - 4)(4q + 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2q^2 - 4q + 4q^2 + 6q}{(4q + 6)(2q - 4)} $ $ = \dfrac{6q^2 + 2q}{(4q + 6)(2q - 4)}$ Expand the denominator: $ = \dfrac{6q^2 + 2q}{8q^2 - 4q - 24}$ Simplify: $ = \dfrac{3q^2 + q}{4q^2 - 2q - 12}$